From an external point P, two tangents PA and PB are drawn to a circle with centre O such that angle APB = 80°. What is the measure of angle AOP (in degrees)?

Introduction / Context:
This circle geometry question deals with two tangents drawn from an external point to a circle and asks for a central angle involving the radius and line to the external point. It tests understanding of tangent properties, isosceles triangles formed by radii, and angle bisector symmetry in such configurations.

Given Data / Assumptions:

• PA and PB are tangents drawn from external point P to a circle with centre O.
• Angle APB, the angle between the tangents, equals 80°.
• We must determine angle AOP.

Concept / Approach:
First, note that OA and OB are radii drawn to the points of tangency A and B, and each radius is perpendicular to the tangent at that point. Thus, angle OAP and angle OBP are right angles (90°). Because tangents from an external point are equal in length (PA = PB), triangle APB is isosceles. The configuration is symmetric about line OP, which acts as the angle bisector of angle APB and also the perpendicular bisector of chord AB. We use this symmetry to find angle APO, angle OAP, and finally angle AOP by applying the triangle angle sum in triangle AOP.

Step-by-Step Solution:
Step 1: Recognise that PA = PB, so triangle APB is isosceles with vertex at P. Step 2: The line OP, joining the centre and external point, is the symmetry axis and bisects angle APB. Step 3: Therefore angle APO = angle OPB = (1/2) × 80° = 40°. Step 4: Radius OA is perpendicular to tangent PA, so angle OAP = 90°. Step 5: Consider triangle AOP. The three angles are angle OAP = 90°, angle APO = 40°, and angle AOP (unknown). Step 6: Use triangle angle sum: angle OAP + angle APO + angle AOP = 180°. Step 7: Substitute known values: 90° + 40° + angle AOP = 180°. Step 8: Simplify: 130° + angle AOP = 180°, so angle AOP = 50°.

Verification / Alternative Check:
Another known relation is that the angle between two tangents (angle APB) is supplementary to the central angle AOB that subtends the chord AB. That is, angle APB = 180° − angle AOB. With angle APB = 80°, we get angle AOB = 100°. Since OP bisects angle AOB as well, angle AOP = 100° / 2 = 50°, confirming the earlier calculation.

Why Other Options Are Wrong:
Values like 40°, 60°, or 70° do not satisfy both the symmetry and the right angle conditions simultaneously. For example, 40° equals the half angle at P, not the central angle at O. A value of 60° or 70° would make the angles in triangle AOP sum to more than or less than 180°, contradicting basic triangle geometry.

Common Pitfalls:
A common mistake is to confuse angle APB with angle AOB or to forget that OP bisects both the angle between the tangents and the corresponding central angle. Students may also overlook the perpendicularity of radius and tangent, forgetting to use the 90° angle at A. Drawing a clear diagram and labelling all known angles helps to avoid these issues.

Final Answer:
The measure of angle AOP is 50°.